a dynamic discrete-continuous choice model of car

Aurélie Glerum EPFL

Emma Frejinger Université de Montréal

Anders Karlström KTH

Muriel Beser Hugosson KTH

Michel Bierlaire EPFL

A DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL OF CAR OWNERSHIP AND USAGE

13th Swiss Transport Research Conference

25th April 2013

OUTLINE 2

• Introduction

• Background and data

• The dynamic discrete-continuous choice modeling framework • Assumptions • Definition of the components • Solving the dynamic programming problem • Model estimation

• Illustration of model application

• Conclusion and future works

INTRODUCTION 3

Aim of the research project: • Model dynamics of car transactions, usage and choice of fuel type in the

Swedish car fleet

• Motivations • Governmental policies:

• Goals of reducing carbon emissions • Technology changes:

• Increase of alternative-fuel vehicles • Changes in the supply

• Company cars: represent important share of new car sales

INTRODUCTION 4

Current literature on car ownership and usage modeling:

• Car ownership models in transportation literature: • Mostly static models:

• Main drawback: do not account for forward-looking behavior • Important aspect to account for since car is a durable good

• Econometric literature: dynamic programming (DP) models + discrete choice models (DCM)

• Recently, dynamic discrete choice models (DDCM) starting to be applied in transportation field (Cirillo and Xu, 2011; Schiraldi, 2011)

• Joint models of car ownership and usage:

• Early references: e.g. duration models and regression techniques for car holding duration and usage (De Jong, 1996)

• Dynamic programming mixed logit (DPMXL) (Schjerning, 2007) used to model car ownership, type of car and usage (Munk-Nielsen, 2012)

• Discrete-continuous model of vehicle choice and usage based on register data (Gillingham, 2012)

Research issues:

• Car are durable goods Need to account for forward-looking behavior of individuals

• Difficulty of modeling a discrete-continuous choice

• Many models focus on individual decisions, but choices regarding

car ownership and usage made at household level

INTRODUCTION 5

Research issues:

• Car are durable goods Need to account for forward-looking behavior of individuals

• Difficulty of modeling a discrete-continuous choice

• Many models focus on individual decisions, but choices regarding

car ownership and usage made at household level

Proposed methodology:

• Attempt to address these issues by applying dynamic discrete-continuous choice model (DDCCM)

• Large register data of all individuals and cars in Sweden

INTRODUCTION 6

BACKGROUND AND DATA 7

Register data of Swedish population and car fleet: • Data from 1998 to 2008

• All individuals

• Individual information: socio-economic information on car holder (age, gender, income, home/work location, employment status/sector, etc.)

• Household information: composition (families with children and married couples)

• All vehicles

• Privately-owned cars, cars from privately-owned company and company cars

• Vehicle characteristics (make, model, fuel consumption, fuel type, age) • Annual mileage from odometer readings • Car bought new or second-hand

Aim of the project: • Model simultaneously car ownership, usage and fuel type.

In details: model simultaneous choice of

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Transaction type

Private/ company car – car c

Fuel type – car c

New/2nd hand – car c

Annual milage – car c

×

×

×

×

# cars

DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL



9

Transaction type


Fuel type – car c



×

×

×

×

# cars


Discrete variables



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Transaction type


Fuel type – car c



×

×

×

×

# cars


Continuous variables

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Motivations for discrete-continous vs discrete model

• Mileage variable(s) are continuous: lose information by

discretizing it.

• In a discrete-continuous approach: If choice of mileage conditional on the discrete choice Reduction of size of the discrete action space


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• Decisions at household level: up to 2 cars in household

• Strategic choice of:

• Transaction • Type(s) of ownership (company vs private car) • Fuel type(s) • Car state(s) (new vs 2nd-hand)

Account for forward-looking behavior of households

• Myopic choice of:

• Annual mileage(s)

• Choice of mileage conditional on choice of discrete variables

DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL ASSUMPTIONS

• Agent: household

• Time step t: year

• State space S

Age – 1st car

Age – 2nd car

Private/ company car – 1st car

Private/ company car – 2nd car

Fuel type – 1st car

Fuel type – 2nd car

DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL 13

DEFINITION OF THE COMPONENTS

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• State space S

Age – 1st car

Age – 2nd car





2-car households

DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL DEFINITION OF THE COMPONENTS

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• State space S

Age – 1st car

Age – 2nd car





1-car households


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• State space S

Age – 1st car

Age – 2nd car





0-car households


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• State space S

Age – 1st car

Age – 2nd car





= 1407 relatively small state space

= 6

= 4

= 4


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• Action space A

Transaction type

Mileage - 2nd car




New/ 2nd hand – 1st car

Mileage – 1st car


New/ 2nd hand – 2nd car


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• Action space A

Transaction type: details


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• Action space A


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• Action space A


Size of the discrete part of the action space assuming: • 3 levels company car • 3 levels fuel type • 2 levels new/2nd hand

Maximum size of action space << for DDCM

DEFINITION OF THE COMPONENTS

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• Transition rule: deterministic rule: each state 𝑠𝑡+1 can be inferred exactly once 𝑠𝑡 and 𝑎𝑡 are known.

Example: If 𝑠𝑡 = [2,1,2,0,0,0] and 𝑎𝑡 = 1,12′000,0,0,0,0,3,0,0 then 𝑠𝑡+1 = 3,1,2,0,3,0 .

2 years

private car

diesel

increase 1

mileage 1st car

company car


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• Instantaneous utility function:


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Deterministic term

Random term

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Assume additive deterministic utility for simplicity (see also Munk-Nielsen, 2012):


Deterministic term

Random term

26




Deterministic term

Random term

Utility for discrete actions

Utility for continuous actions

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Deterministic term

Random term

Utility for discrete actions

Utility for continuous actions

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• Instantaneous utility function

• Utility for continuous actions: Constant elasticity of substitution (CES) utility function:

• Captures substitution patterns between the choice of both annual driving distances

• ρ is elasticity of substitution

• Randomness introduced in

• 𝑥𝑡 contains price of fuel, car consumption and other socio-economic characteristics


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1. Finding the optimal value(s) of annual mileage conditional on the discrete choices

2. Solving the Bellman equation

DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL SOLVING THE DYNAMIC PROGRAMMING PROBLEM

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Finding the optimal value(s) of mileage

• Maximization of the continuous utility:

s.t. • Find analytical solutions: and

• Optimal continuous utility


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Solving the Bellman equation

• Logsum formula used in the completely discrete case (DDCM)

(Aguirregabiria and Mira, 2010; Cirillo and Xu, 2011)

• Logsum can be applied here given the key assumptions: • Choice of mileage(s) is conditional on discrete actions • Choice of mileage(s) is myopic

• Iterate on Bellman equation to find integrated value function 𝑽�


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• Parameters obtained by maximizing likelihood:

• Optimization algorithm is Rust’s nested fixed point algorithm (NFXP) (Rust, 1987): • Outer optimization algorithm: search algorithm to obtain parameters

maximizing likelihood

• Inner value iteration algorithm: solves the DP problem for each parameter trial

• Plan to investigate variants of NFXP to speed up computational time

MODEL ESTIMATION DYNAMIC DISCRETE-CONTINUOUS CHOICE MODEL

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Assumptions for the example:

• Size state space = 651 • Max age = 3 • Company car levels = 3 • Number of fuel types = 3

• Size action space = max 745

• Number of transaction types = 9 • Number of state levels (new/old) = 2

• Utility function contains:

• Transaction cost 𝜏 • Transaction-dependent parameters for age of oldest car

• Parameters of DP problem: • Discount factor β = 0.7 • Stopping criterion 𝜀 = 0.01

postulated

ILLUSTRATION OF MODEL APPLICATION

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Program:

• Code in C++

• 2 minutes on 20-core server Graph:

• 𝑽� vs age of oldest of cars in

household fleet

ILLUSTRATION OF MODEL APPLICATION

35 ILLUSTRATION OF MODEL APPLICATION


Instantaneous utility


Expected discounted utility

CONCLUSION AND FUTURE WORKS 38

Conclusion:

• Methodology to model choice of car ownership and usage dynamically

• Example of application shows feasibility of approach

Next steps: • Exploratory analysis to specify instantaneous utility

• Model estimation on small sample of synthetic data

• Model estimation on register data

• Scenario testing: • Validation of policy measures taken during the years available in the data

• Test policy measures that are planned to be applied in future years

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Thanks!

a dynamic discrete-continuous choice model of car

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